Conditional probability by joint distribution

Question

I have read the following equation:
$$P(E_1|H)*P(E_2|H)*P(H) = P(E_1,E_2,H) $$
I don't understand why this equation is true. I know that
$$P(E_1|H)*P(H) = P(E_1,H)$$
thus: $$P(E_1|H)*P(E_2|H)*P(H)=P(E_1,H)*P(E_2|H)$$ but why: $$P(E_1|H)*P(E_2|H)*P(H) = P(E_1,E_2,H) $$ As a extra information it is also given that:
$$P(E_1|H,E_2) = P(E_1|H)$$

Answer 1

We are given that $$P(E_1|H,E_2)=P(E_1|H)$$

hence,

$$\frac{P(E_1,H,E_2)}{P(H,E_2)}=P(E_1|H)$$

$$P(E_1,H,E_2)=P(E_1|H)P(H,E_2)$$

but we have $P(H,E_2)=P(H)P(E_2|H)$, hence the result.